Not, you don't need to know Java EE to learn Spring. Spring has support for just about all Java environments, even the most basic Java apps.
I don't want to confuse that with Spring Web or related components, however. The Spring Framework supports a large number of modules of different types, each of which is optional, depending on what you want to do. For example, there's Spring Data, which deals with different types of persistent data storage including both SQL and non-SQL databases. Within Spring Data you can find Spring JPA, which facilitates the use of the Java Persistence Architecture, which is defined as an independent subset of the JEE EJB3 data standard.
Spring Web provides a Model/View/Controller (MVC) framework for designing web applications, but it's not required. Indeed, my favorite setup for web applications is to use JavaServer Faces for the GUI front-end, Spring JPA for the backend, and other Spring services (such as email support) if and as needed. It all works together quite nicely.
Now if you specifically want to create web applications using Spring, the one thing you do want to be familiar with is how the HTTP protocol works - how requests and responses apply, what JEE HttpSessions are and how JEE security works. Those are key whether your use Spring Web, JSF, or any other web framework.
I'm going to be a "small government" candidate. I'll be the government. Just me. No one else.